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The area bounded by the curves y = sin x, y = cosx and y-axis in first quadrant is
- a)(√2 − 1) sq.unit
- b)√2 sq.unit
- c)(√2 + 1) sq.unit
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
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The area bounded by the curves y = sin x, y = cosxand y-axis in first ...
The area bounded by the curves y = sin x, y = cos x, and the y-axis in the first quadrant is represented by the shaded region in the graph below:
[graph here]
To find the area, we need to find the x-values where the curves intersect.
The curves y = sin x and y = cos x intersect when sin x = cos x. We can solve this equation by setting sin x = cos x:
sin x = cos x
Dividing both sides by cos x:
tan x = 1
Taking the inverse tangent of both sides:
x = π/4
Therefore, the curves intersect at x = π/4.
To find the area, we integrate the function y = sin x - cos x from x = 0 to x = π/4:
∫[0,π/4] (sin x - cos x) dx
Integrating sin x gives -cos x, and integrating cos x gives sin x. Evaluating the integral from 0 to π/4:
[-cos x + sin x] [0,π/4]
= (-cos(π/4) + sin(π/4)) - (-cos(0) + sin(0))
= (-sqrt(2)/2 + sqrt(2)/2) - (-1 + 0)
= sqrt(2)/2 + 1
Therefore, the area bounded by the curves y = sin x, y = cos x, and the y-axis in the first quadrant is sqrt(2)/2 + 1.
[graph here]
To find the area, we need to find the x-values where the curves intersect.
The curves y = sin x and y = cos x intersect when sin x = cos x. We can solve this equation by setting sin x = cos x:
sin x = cos x
Dividing both sides by cos x:
tan x = 1
Taking the inverse tangent of both sides:
x = π/4
Therefore, the curves intersect at x = π/4.
To find the area, we integrate the function y = sin x - cos x from x = 0 to x = π/4:
∫[0,π/4] (sin x - cos x) dx
Integrating sin x gives -cos x, and integrating cos x gives sin x. Evaluating the integral from 0 to π/4:
[-cos x + sin x] [0,π/4]
= (-cos(π/4) + sin(π/4)) - (-cos(0) + sin(0))
= (-sqrt(2)/2 + sqrt(2)/2) - (-1 + 0)
= sqrt(2)/2 + 1
Therefore, the area bounded by the curves y = sin x, y = cos x, and the y-axis in the first quadrant is sqrt(2)/2 + 1.
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Community Answer
The area bounded by the curves y = sin x, y = cosxand y-axis in first ...
In first quadrant sin x and cos x meet at x = π/4.
The required area is as shown in the diagram.
So, the required area
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The area bounded by the curves y = sin x, y = cosxand y-axis in first quadrant isa)(√2 − 1) sq.unitb)√2 sq.unitc)(√2 + 1) sq.unitd)none of theseCorrect answer is option 'A'. Can you explain this answer?
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